Global Gene Regulation in Metabolic Networks

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Global Gene Regulation in Metabolic Networks

Diego Oyarzun, Madalena Chaves

To cite this version:

Diego Oyarzun, Madalena Chaves. Global Gene Regulation in Metabolic Networks. [Research Report]

RR-7468, INRIA. 2010, pp.15. �inria-00541112�

a p p o r t

d e r e c h e r c h e

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Global Gene Regulation in Metabolic Networks

Diego A. Oyarzún — Madalena Chaves

N° 7468

Centre de recherche INRIA Sophia Antipolis – Méditerranée 2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex

Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65

Global Gene Regulation in Metabolic Networks

Diego A. Oyarzún

_{, Madalena Chaves}

Thème : Observation, modélisation et commande pour le vivant Équipe-Projet Comore

Rapport de recherche n° 7468  November 2010  15 pages

Abstract: Genetic feedback is one of the mechanisms that enables metabolic adaptations to environmental changes. The stable equilibria of these feedback circuits determine the observable metabolic phenotypes. We consider an un-branched metabolic network with one metabolite acting as a global regulator of enzyme expression. Under switch-like regulation and exploiting the time scale separation between metabolic and genetic dynamics, we develop geometric cri-teria to characterize the equilibria of a given network. These results can be used to detect mono- and bistability in terms of the gene regulation parameters for any combination of activation and repression loops.

Key-words: kinetic modeling and control of biological systems, metabolic networks, genetic networks

The rst author was supported by a NBIPI Career Enhancement and Mobility Fellowship (co-funded by Marie Curie Actions, the Irish Higher Education Authority Programme for Third Level Institutions Cycle 4 and the Italian National Research Council). The second author was supported in part by the INRIA-INSERM project ColAge.

∗_{Hamilton Institute, National University of Ireland, Maynooth, Co. Kildare, Ireland} (e-mail: diego.oyarzun@nuim.ie).

†_{COMORE, INRIA Sophia Antipolis, 2004 route des Lucioles, BP 93, 06902 Sophia} An-tipolis, France (e-mail: madalena.chaves@inria.fr)

Régulation Génétique Globale de Réseaux

Métaboliques

Résumé : Les réactions métaboliques sont régulées par des enzymes mais, en même temps, les gènes codant pour ces enzymes peuvent être régulées par les produits des réseaux métaboliques. Ce mécanisme de régulation en boucle fermée permet l'adaptation d'un système métabolique aux changements envi-ronnementaux. Nous étudions un système composé d'une chaine métabolique couplée à un réseau génétique, où un des métabolites joue le rôle de régulateur global de l'expression enzymatique. Le métabolite régulateur peut être soit ré-presseur soit activateur de chaque gène. Sous l'hypothèse que l'échelle de temps du système métabolique est plus rapide que celle du réseau génétique, le sys-tème composé peut être simplié. Dans ce cas, nous développons des critères géométriques pour caractériser les équilibres du système et analyser la mono-ou bi- stabilité du système en termes des paramètres.

Mots-clés : modélisation et contrôle de systèmes biologiques, réseaux métaboliques, réseaux génétiques

Regulation of Metabolic Networks 3

1 Introduction

Metabolism and gene expression are two fundamental levels of cellular regula-tion. Gene expression can impact metabolic levels through changes in enzyme concentration and, conversely, metabolic species can inuence gene transcrip-tion and hence modulate enzyme synthesis [9]. These two levels have specic functions and properties, and it remains a challenge to characterize the inter-connections between them. The congurations of metabolic-genetic interactions can lead to a diverse range of dynamic behaviors, each one of which denes a specic metabolic phenotype. Our understanding of natural regulatory circuits is important not only for revealing the design principles that underlie observed metabolic dynamics, but also for our ability to design synthetic circuits that enable new phenotypes [1].

A specic phenotype depends, among others, on: the regulatory topology (i.e. which metabolites regulate which enzymes), logic (i.e. activation or repres-sion), and the sensitivity of regulation (i.e. graded or switch-like regulation). In this paper we present a mathematical analysis of these properties by studying the interconnection between an unbranched metabolic network and a one-to-all genetic control circuit. The gene circuit implements a form of global control in the sense that one metabolic species modulates the activity of all enzymes.

The model integrates the classical kinetic equations to represent metabolite dynamics and piecewise ane (PA) systems to represent gene regulation (Sec-tion 2). To analyze the model we consider that metabolic reac(Sec-tions happen in a much faster time scale when compared to gene transcription or translation [1]. PA models [2] provide a convenient way of encoding switch-like regulation with relatively few parameters (only expression rates and regulatory thresholds), whereas the time scale separation allows for an approximation of the metabolic dynamics by a system of algebraic equations. This approximation ultimately leads to a reduction of the model to a 2-dimensional PA system dened in conic domains (Section 3). Solutions of these type of systems can be obtained by dening the system as a dierential inclusion and using a construction due to [5]. The conic geometry introduces new features in the solutions at the switching domains, and we characterize the existence and stability of the possible sliding motion and singular equilibria (Section 4). This analysis is applied to study a particular conguration of the globally regulated metabolic chain (Section 5).

2 Global genetic control

We consider an unbranched metabolic network with one metabolite as a global regulator of enzyme expression. A schematic diagram of such class of networks is shown in Figure 1, where si denotes the concentration of the ith metabolite

and viis the rate of the ithreaction (catalyzed by an enzyme with concentration

ei).

As a way of accounting for the mass exchange between the network and its environment, we assume that the metabolic substrate s0 is constant. For the

sake of generality, in this paper we deal with networks of n metabolites and n + 1 enzymes regulated by a single global regulator s`−1 (` > 1). The rate of

4 Oyarzún & Chaves

e

s

v

e

s

s

v

e

e

v

v

s

Figure 1: Global gene regulation for an unbranched network.

change of both metabolite and enzyme concentrations can be described by ˙si= vi(si−1, ei) − vi+1(si, ei+1), (1)

˙ei= κ0i + κ 1 iσi(s`−1, θi) − γiei. (2) where κ0 i, κ 1

i, θi, γiare positive parameters. The metabolic model (1) arises from

the mass balance between the reactions that produce and consume si, whereas

the model for the enzyme concentrations (2) comes from the balance between protein synthesis and degradation (modeled as a linear process). The constant κ0_i represents a basal expression level of protein ei, whereas κ1i and the functions

σi model the eect of the global regulator on the synthesis rates.

The regulatory function σi(s`−1, θi)represents the lumped eect of gene

ex-pression control by a transcription factor, together with its interaction with the regulator s`−1. To account for the typical switch-like nature of transcriptional

regulation, σi(x, θ)is assumed to be a step function; depending on whether gene

expression is activated or repressed by s`−1, we assign σi = σ+ or σi = σ− =

1 − σ+_{, respectively, with}

σ+(x, θ) = (

0, x < θ

1, x > θ. (3) This class of regulatory functions is widely used used in the analysis of genetic networks [10, 3] and was rst suggested in [6]. In the sequel we will not pre-suppose a specic form of the enzyme kinetics; instead to keep the analysis as general as possible, we make the following assumption on the enzyme kinetics. Assumption 1. The metabolic reaction rates are linear in the enzyme concen-trations and non-decreasing functions of the metabolite concenconcen-trations, so that the rate functions are written as

vi(si−1, ei) = gi(si−1)ei, (4)

where gi is the enzyme turnover rate (i.e. the reaction rate per unit of enzyme

concentration) and saties

∂gi(si−1)

∂si−1

≥ 0. (5)

The monotonicity condition in (5) accounts for a broad class of saturable enzyme kinetics that includes, in particular, Michaelis-Menten and Hill kinetics [4]. In the rest of the paper we aim at characterizing the equilibria of the feedback system in (1)(2).

Regulation of Metabolic Networks 5

3 Time scale separation

3.1 Quasi steady state approximation

Metabolic dynamics operate in a much shorter time scale than their genetic counterpart [1]. This property allows the approximation of the nonlinear dy-namics in (1) by an algebraic relationship between the enzymes and metabolite concentrations. If the metabolites are assumed to be in quasi steady state (QSS) with respect to the enzyme concentrations, then we set ˙si(t) = 0for all t ≥ 0 to

obtain

gi+1(si(t)) = gi(si−1(t))

ei(t)

ei+1(t)

, (6)

Equation (6) holds for every i = 1, 2, . . . , n and hence it is equivalent to

gi(si−1(t)) = g1(s0)

e1(t)

ei(t)

. (7)

The trajectory of the global regulator can be then computed by solving the equation

g`(s`−1(t)) = g1(s0)

e1(t)

e`(t)

. (8)

If the function gi does not saturate, a nonnegative solution of (7) exists at all

times. However, for saturable functions one must guarantee that this is true by stating appropriate assumptions on the parameters (for instance, of the form κ0_i/γi≥ (g1(s0)/ max(gi))(κ01+ κ11)/γ1). For space reasons we will assume that

solutions of (7) do exist and omit the details. A key aspect of this approximation is that the solution of (1)(2) depends only on two proteins. The dynamics of the complete feedback system can thus be fully characterized by analyzing the 2-dimensional phase plane of the dierential equations

˙e1= κ01+ κ11σ1(s`−1, θ1) − γ1e1, ˙e`= κ0`+ κ 1 `σ`(s`−1, θ`) − γ`e`, (9) subject to s`−1 satisfying (8).

3.2 Equivalent piecewise ane system in conic domains

The algebraic equation in (8) can be interpreted as a mapping from R≥0 to R2≥0,

whereby each value of the regulator s`−1 maps into a half-line in the (e1, e`)

plane. Moreover, as a consequence of the monotonicity of g`, the partition

of R≥0 induced by the thresholds can be mapped into a partition of R2≥0: if

s`−1< θi then

g`(s`−1) < g`(θi), (10)

which combined with (8) yields

e`> βie1, (11)

6 Oyarzún & Chaves

with βi= g1(s0)/g`(θi). The relation in (11) denes a cone in the x = (e1, e`)

plane

Di=x ∈ R2≥0 : x2> βix1 , (12)

and we dene its complementary cone as ¯Di = R2_≥0\ (Di∪ Si) with Si the

half-line

Si=x ∈ R2≥0: x2= βix1 . (13)

The half-line Si is a subset of the (e1, e`)plane where the regulator reaches the

switching threshold θi. The dynamics of the reduced system in (9) depend on

the value of s`−1 with respect to the thresholds θ1 and θ`. Assume, without

loss of generality, that θ1 < θ` (the problem can be treated analogously in the

case θ1> θ`, and the case θ1= θ` is treated as in Section 4). With the previous

denitions we can establish the following relations s`−1< θ1⇐⇒ x ∈ R1,

θ1< s`−1< θ`⇐⇒ x ∈ R1`,

s`−1> θ`⇐⇒ x ∈ R`,

(14)

where R1= D1, R1`= ¯D1∩ D` and R`= ¯D`. In the sequel we refer to Si as a

switching domain, whereas the cones Rj are called regular domains (see [2] for

detailed denitions). The system in (9) is equivalent to a piecewise ane (PA) system [8] in conic domains

˙ x = h(x) − Γx, (15) where h(x) =      h1 x ∈ R1 h1` <sub>x ∈ R</sub> 1` h` x ∈ R` , Γ =γ1 0 0 γ`  . (16)

The vectors h1_{, h}1` <sub>and h</sub>` _{are constant and their values depend on whether}

s`−1 activates or represses the expression of enzymes e1 and e`. For example,

in the case of repression (i.e. σ1= σ− and σ`= σ−) we have

h1=κ 0 1+ κ11 κ0 `+ κ1`  , h1`=  κ0 1 κ0 `+ κ1`  , h`=κ 0 1 κ0 `  . (17) These vectors determine the location of the focal points of the PA system, dened as

φ1= Γ−1h1, φ1`= Γ−1h1`, φ`= Γ−1h`. (18) The conic partition of the (e1, e`) phase plane and the focal points are shown

in Figure 2. For any x(t0)in a regular domain, e.g. x(t0) ∈ R1, the right-hand

side of (20) is well dened and its solution satises a standard ane dierential equation, that is

x(t) = φ1+ eΓ(t0−t) _x(t

0) − φ1
, t ≥ t0 (19)

so that x(t) monotonically approaches φ1_{, possibly reaching the switching}

do-main S1, where the vector eld of (20) is not dened, and thus a specialized

analysis is required. As we shall see in the next section, the location of the focal points plays a major role in the dynamics of (15).

Regulation of Metabolic Networks 7

e

φ

e

S

S

φ

φ

R

R

R

Figure 2: Conic partition of the state space and focal points when both e1 and

e` are repressed by metabolite s`−1.

4 Piecewise ane systems in cones

To understand the dynamics of the PA system in (15), in this section we rst study the solutions of a 2-dimensional PA system dened only in two cones (as opposed to three cones, cf. Figure 2). Consider the half-line S = x ∈ R2

≥0: x2= βx1

with β > 0, and the PA system

˙ x = ( f (x) x ∈ Df g(x) x ∈ Dg , (20) where Df = x ∈ R2≥0: x2> βx1

and Dg = R2≥0\ (S ∪ Df).We denote the

generating vector of S as η = 1 βT

; the vector elds f(x) = f1(x) f2(x)

T and g(x) = f1(x) f2(x)

T

are ane and given by

f (x) = hf− Γx, g(x) = hg− Γx, (21) where hf _{and h}g _{are entrywise positive vectors, and Γ = diag {γ}

1, γ2} > 0 .

The focal points of f(x) and g(x) are given by φf _{= Γ}−1_hf _{and φ}g _{= Γ}−1_hg_,

respectively, and are assumed to satisfy φf_{, φ}g _{∈ S/ . Solutions of dierential}

equations with discontinuous vector elds are typically characterized with a construction due to Filippov [5]. This method proceeds by extending (20) to a dierential inclusion

˙

x ∈ H(x), ∀x ∈ S, (22) where H(x) is a set-valued function dened as the closed convex hull of f(x) and g(x), i.e.

H(x) =_{z ∈ R}2: z = αf (x) + (1 − α)g(x), 0 ≤ α ≤ 1 .

The solutions of (22) are understood in the following sense (see also [7, 2] for more details).

Denition 4.1 or a given ρ0, a solution of (22) in [0, T ] is an absolutely

con-tinuous function ρ : [0, T ] → R2

≥0 such that ρ(0) = ρ0 and ˙ρ(t) ∈ H(ρ(t)) for

almost all t ∈ [0, T ].

8 Oyarzún & Chaves

4.1 Solutions in the switching domain

Depending on the directions of the vector elds f(x) and g(x), Filippov's con-struction may not allow for uniqueness of solutions in the switching domains [5]. When uniqueness can be guaranteed, then solutions of (22) can: (a) cross to a regular domain or, (b) slide along the switching surface S. Roughly speaking, case (a) occurs when f(x) and g(x) point in similar directions in a vicinity of S, so that the vectors in H(x) point toward a regular domain irrespective of α. In case (b) both vector elds point towards the switching domain, so that one can nd a unique value of α such that H(x) points in the direction of S (in this case we say that the solution exhibits stable sliding motion in S). Uniqueness of solutions is lost when both vector elds point away from the switching do-main, in which case solutions starting at S cannot be uniquely dened and any small perturbation will drive x away from S (referred to as an unstable sliding motion).

Next we identify the above scenarios for the PA system dened in (20). To that end we dene the sets

Ω−_f = {x ∈ Df∪ S : f2(x) − βf1(x) ≤ 0} ,

Ω+_f = {x ∈ Df∪ S : f2(x) − βf1(x) > 0} ,

Ω−_g = {x ∈ Dg∪ S : g2(x) − βg1(x) ≤ 0} ,

Ω+g = {x ∈ Dg∪ S : g2(x) − βg1(x) > 0} .

(23)

Lemma 4.1 [Crossings between regular domains] The solutions of (20) cross from Dg to Df in the set

Lgf = Ω+f ∩ Ω +

g ⊆ S, (24)

and cross from Df to Dgin the set

Lf g= Ω−f ∩ Ω −

g ⊆ S, (25)

Proof: We only prove the rst case (i.e. that if the solution reaches Lgf dened

in (24), then it crosses from Dg to Df); the converse case follows analogously.

From Filippov's construction, the vector eld in S has the form

˙

x =αf1(x) + (1 − α)g1(x) αf2(x) + (1 − α)g2(x)



, (26)

with 0 ≤ α ≤ 1. By the denition of Ω+ f and Ω

+

g, for any point x ∈ Lgf we have

that

αf2(x) + (1 − α)g2(x) > αβf1(x) + (1 − α)βg1(x), (27)

> β (αf1(x) + (1 − α)g1(x)) ,

which implies that the vector eld points to Df for any 0 ≤ α ≤ 1. In other

words, the set H(x) in (22) is fully contained in the regular domain Df and

hence the trajectory crosses the switching domain. 2 Lemma 4.2 [Stable sliding motion] The solutions of (20) exhibit stable sliding motion in the set

Ls= Ω−f ∩ Ω +

g ⊆ S. (28)

Regulation of Metabolic Networks 9

Proof: We rst prove by contradiction that for all x(t0) ∈ Ls the vector elds

are such that x(t) cannot leave the switching domain in an interval (t0, t0+ ∆ ]

. Dene the absolutely continuous function z : [t0, t0+ ∆] → R,

z(t) = x2(t) − βx1(t).

(29)

Suppose that there exists ∆ > 0 such that z(t) > 0 for t ∈ (t0, t0+ ∆ ]. If

x(t0) ∈ S we have that z(t0) = 0, so by continuity it must be that ˙z > 0 for

t ∈ ( t0, t0+ ∆0]and some 0 < ∆0≤ ∆. In addition, from the denition of the

PA system in (20), if z(t) > 0 then ˙x = f(x) for t ∈ (t0, t0+ ∆0] and so

˙

z = f2(x) − βf1(x), for t ∈ ( t0, t0+ ∆0] . (30)

However, the right-hand side of (30) is continuous in t, and when x(t0) ∈ Ls it

follows that ˙z ≤ 0 for t ∈ (t0, t0+ ∆0], which is a contradiction. The converse

argument can be used to show that z(t) < 0 for t ∈ (t0, t0+ ∆ ] leads to a

contradiction. We thus conclude that z(t) = 0 for t ∈ [t0, t0+ ∆], and so

x(t) ∈ Ls for t ∈ [t0, t0+ ∆].

The proof follows by checking that the vector elds for x ∈ Lsare compatible

with Filippov's construction, see [7]. If there is sliding motion in Ls, then there

exists ∆ > 0 such that ˙

z = 0, for t ∈ [t0, t0+ ∆] . (31)

Since x(t) must be a solution in Filippov's sense for t ∈ [t0, t0+ ∆], then there

must exist 0 ≤ α ≤ 1 such that ˙

x = αf (x) + (1 − α)g(x), for t ∈ [t0, t0+ ∆] (32)

Combining (31) and (32) we get 0 = ˙x2− β ˙x1,

= αf2+ (1 − α)g2− β (αf1+ (1 − α)g1) ,

= α(f2− βf1) + (1 − α)(g2− βg1), for t ∈ [t0, t0+ ∆] . (33)

Solving for α in (33) gives

α(x) = g2(x) − βg1(x)

(g2(x) − βg1(x)) − (f2(x) − βf1(x))

, (34) For x ∈ Ls it holds that (f2− βf1) ≤ 0 and (g2− βg1) > 0, therefore α(x) is

unique for all x ∈ Ls and satises 0 ≤ α(x) ≤ 1. 2

Lemma 4.3 [Unstable sliding motion] The solutions of (20) cannot be uniquely dened in the set

Ls¯= Ω+f ∩ Ω −

g ⊆ S. (35)

Proof: Consider the function z(t) dened in (29). As opposed to the proof of Lemma 4.2, in this case it can be shown that for x(t0) ∈ L¯sboth z(t) > 0 and

z(t) < 0for t ∈ (t0, t0+ ∆ ] are possible solutions. Note that another possible

solution can be dened by picking α as in (34) so that x(t) slides along S. 2

10 Oyarzún & Chaves

4.2 Geometric representation

The results of last section dene a partition of the switching domain into regions for crossings and sliding motions that has a convenient geometric representa-tion. An alternative denition of the sets in (23) can be constructed as follows: dene the normal vector of S as η⊥ _{= −β 1}T_{, then x ∈ Ω}−

f if and only if

hf (x), η⊥_{i ≤ 0, which after substitution of f(x) = h}f _{− Γxbecomes}

hΓ(x − φf), η⊥i ≥ 0. (36) Analogous denitions can be constructed for the other sets in (23). The bound-ary between Ω− f and Ω + f is the half-line Cf =x ∈ R2≥0: hΓ(x − φ f ), η⊥i = 0 , (37) whereas the boundary between Ω−

g and Ω+g is

Cg=x ∈ R2≥0: hΓ(x − φg), η⊥i = 0 . (38)

The focal points satisfy φf _{∈ C}

f and φg ∈ Cg, respectively. Moreover, Cf

and Cg are parallel and thus the sets Ls and Ls¯correspond to the intersection

between S and the band generated by Cf and Cg. Figure 3 illustrates this

idea; a necessary condition for sliding motion (stable or unstable) is that the band between Cf and Cg intersects S in R2≥0. Moreover, inspection of Figure

3 suggests that the partition precludes the existence of stable and unstable sliding motion in the same switching domain (i.e. at least one of the sets Lsand

L¯s is empty). Next we provide a geometric condition for characterizing these

scenarios. Lemma 4.4 Let ϑ1= ∠ Γ φf− φg
, η⊥
. If ϑ1∈ h −π, −π 2  ∪π 2, π i , (39)

unstable sliding motion cannot exist in S. Conversely, if ϑ1∈  −π 2, π 2  , (40)

stable sliding motion cannot exist in S.

Proof: Cf and Cg intercept the vertical axis of R2_≥0 at pf = γ2−1hΓφf, η⊥i

and pg _{= γ}−1 2 hΓφ

g_{, η}⊥_{i, respectively. Whether L}

s = ∅or L¯s = ∅ depends on

sgn pf− pg
_{= sgnhΓ φ}f_{− φ}g
_{, η}⊥_{i (see Figure 3), which leads to the angle}

conditions in (39)(40). 2

4.3 Equilibria

The focal points are locally stable equilibria of (20) provided that they belong to their respective cone, i.e. φf _{∈ D}

for φg∈ Dg(stability follows from γ1, γ2> 0).

However, as suggested by Figure 3, it is possible that the trajectories reach an equilibrium that lies in the switching domain. This kind of equilibrium is sometimes termed singular equilibrium [2] and must be understood in Filippov's

Regulation of Metabolic Networks 11 x1 φf Ω−_f Ω+g Cg Ω _S + f φg Cf Ω−g Ls x2 Ls¯ x2 S x1 Cf φg φf Ω+_f Ω−_f Cg Ω+g Ω−g

Figure 3: Switching domain with stable (left) and unstable (right) sliding mo-tion.

sense (that is, at a singular equilibrium the convex hull H(x) in (22) contains the origin). Depending on the location of the focal points, a singular equilibrium may exist in Ls; we characterize this statement more precisely in the next

lemma.

Lemma 4.5 [Singular equilibrium] Assume that Ls 6= ∅; let Lφ be the line

containing φf _{and φ}g _{and ϑ}

2= ∠ φf− φg, η⊥
. The point

φs= Ls∩ Lφ, (41)

is a singular equilibrium of (20). Moreover, if ϑ2∈ h −π, −π 2  ∪ π 2, π i , (42)

then φs is locally stable, and if

ϑ2∈  −π 2, π 2  , (43) then φs is unstable.

Proof: The proof follows by looking at the form of the vector eld along S when solutions are dened with Filippov's method. When x ∈ Lsthe solution satises

˙

x = αf (x) + (1 − α)g(x), (44) with α = α(x) given in (34). Substituting α(x) in (44) we get

˙ x = Af g(x)η (g2(x) − βg1(x)) − (f2(x) − βf1(x)) , (45) where Af g(x)is given by Af g(x) = γ1γ2 n xTP φf− φg
+ φf T_{P φ}go_{, (46)} with P =  0₋₁ 1₀ 

so that xT_{P x = 0 for all x ∈ R}2_{. A point φ}

s ∈ Ls is a

singular equilibrium of (20) if it satises Af g(φs) = 0. The equation Af g(x) = 0

is satised by both focal points, i.e. Af g φf
= Af g(φg) = 0, and so the curve

Lφ=x ∈ R2≥0: Af g(x) = 0 , (47)

12 Oyarzún & Chaves

is the line containing both focal points. We thus conclude that any singular equilibrium must be located at φs = Lφ∩ Ls. The stability of φs follows by

examining the direction of the vector eld in (45). We know that

(g2(x) − βg1(x)) − (f2(x) − βf1(x)) > 0, (48)

for all x ∈ Ls, and hence the direction of the right-hand side of (45) depends

only on the sign of Af g(x)along Ls. The function Af g(x) evaluated along Ls

(i.e. when x = x1· η) denes a line

Af g(x)|_x∈L s = γ1γ2 n hφf_{− φ}g_{, η}⊥_ix 1+ φf TP φg o , (49) with slope ∂ ∂x1 Af g(x)|_x∈L s = γ1γ2hφ f_{− φ}g_{, η}⊥_{i. (50)}

Note that the line in (49) is transversal to the line Lφ and they intersect at φs

(because Af g(φs) = 0and φs∈ Ls). Therefore Af g(x) changes sign at x = φs,

so the local stability of φs depends on the sign of the slope in (50); namely

φsis

(

stable if hφf_{− φ}g_{, η}⊥_{i < 0}

unstable if hφf− φg_{, η}⊥_{i > 0}, (51)

which are equivalent to the angle conditions in (42)(43) (note that hφf ₋

φg_{, η}⊥_{i 6= 0since φ}f_{, φ}g_{∈ S/ by assumption). 2}

The result in Lemma 4.5 provides a geometric condition to check the exis-tence of singular equilibrium points for the PA system (20). These equilibria can be locally stable or unstable, a property that can be graphically checked with a simple angle condition; the stable and unstable cases are illustrated in Figure 4(a) and Figure 4(b), respectively. The PA system can also lack a singu-lar equilibrium, in which case Ls∩ Lφ= ∅and the solutions may exhibit sliding

motion in Lsbut eventually escape to one of the regular domains; this scenario

is shown in Figure 4(c). (a) φf φg φs η⊥ Ls Cg ϑ2 S Cf (b) φ g Cf S Ls φs φf Cg η⊥ ϑ2 (c) φg Cg φf Cf S Ls

Figure 4: Singular equilibrium for the PA system in (20).

Regulation of Metabolic Networks 13

5 Equilibria of genetic control circuit

In this section we return to the original problem of determining the metabolite and enzyme equilibria of the system in (1)(2), after its reduction to a PA system dened in three conic domains, see (15). If θ1= θ`, then the region R1`

is empty, and the equilibria can be obtained directly with the analysis in Section 4. In the case θ1 6= θ`, the idea is to use Lemmas 4.4 and 4.5 by splitting the

three cone case into a pair of two cone problems: one for the switching domain S1with the focal points φ1and φ1`, and another one for S`with the focal points

φ1` <sub>and φ</sub>`_.

As discussed in Section 3.2, the location of the focal points depends on the regulatory logic in (9), which is comprised in the functions σ1and σ`. If θ16= θ`

and regardless of the particular regulatory logic, one pair of focal points will share the same vertical coordinate and another pair will share the horizontal coordinate. We can then use our results to show that the system can have one or two stable equilibria. Due to space constraints we only focus in the case of Figure 2, with θ1< θ`.

Lemma 5.1 [Equilibrium of genetic circuit] Consider the PA system in (15) with σ1 = σ− and σ` = σ−, θ1 < θ`, and the focal points

located as in Figure 2. Then, (15) has only one equilibrium point φ1

s, which is

singular, locally stable, and located at

φ1_s= Lφ∩ S1, (52)

where Lφis the line containing φ1 and φ1`.

Proof: In Figure 2 we see that the focal points lie outside their corresponding regular domains, so they cannot be equilibria of (15). The only option then is to look for singular equilibria in S1or S`. Denote the normal vectors to S1 and

S` as η⊥1 and η⊥` , respectively. We proceed by cases:

ˆ φf _{= φ}1` <sub>and φ</sub>g<sub>= φ</sub>` _{are aligned vertically, and hence}

−π

2 ≤ ∠ Γ φ

1`<sub>− φ</sub>`
_{, η}⊥

`
≤ 0, (53)

for all γ1, γ2 > 0; from Lemma 4.4, (53) implies that S` cannot contain

any stable sliding motion.

ˆ φf _{= φ}1 _{and φ}g_{= φ}1` _{are aligned horizontally, and thus we have}

−π ≤ ∠ Γ φ1_{− φ}1`
_{, η}⊥ 1
≤ −

π

2, (54) for all γ1, γ2 > 0; from Lemma 4.4 (54) implies that S1 can contain a

region for stable sliding motion; the existence, location and stability of a singular equilibrium in S1follow directly from Lemma 4.5 and inspection

of Figure 2. 2

The equilibrium enzyme concentrations for the feedback system in (1)(2) can be directly obtained with Lemma 5.1. Assuming that θ1 < θ` and given

14 Oyarzún & Chaves

that φ1

s∈ S1is a singular equilibrium, we have ¯s`−1= θ1and hence (see Figure

2) ¯ e1= 1 β1 ¯ e`, e¯`= κ0 `+ κ1` γ` . (55)

The remaining equilibrium enzymes can be computed from (2) as

¯ ei= κ0 i + κ1iσi(θ1, θi) γi , i 6= 1, (56) where the threshold are assumed to satisfy θi 6= θ1 for i 6= 1. The equilibrium

metabolites can be computed by solving the algebraic equation (cf. (7))

gi(¯si−1) = g1(s0) ¯ e1 ¯ ei , i > 1, i 6= `. (57) As an illustrative example, in Figure 5 we plot the (e1, e`) phase plane and

the metabolite trajectories for the system in (1)(2), together with its QSS approximation in (9). The simulations in Figure 5 were obtained for a system of length n = 2 with σ1 = σ−, σ` = σ−, substrate s0 = 1, Michaelis-Menten

kinetics (gi = kcat isi−1/(Km i + si−1), with kcat i = 2 · 102 and Km i = 5,

i = 1, 2, 3), regulatory parameters θi = {0.5, 1, 1.5}, κ0i = {0.08, 0.15, 0.3},

κ1

i = 10κ0i, and degradation rates γi= {0.25, 0.9, 0.8}.

0 1 2 3 4 0 1 2 3 4 e₁ e3 Without QSS With QSS 0 5 10 0 1 2 3 Metabolite s 1 0 5 10 0 1 2 3 Time Metabolite s 2

Figure 5: Numerical example; eect of QSS approximation.

6 Discussion & outlook

The analysis carried out in this paper shows how genetic regulation of enzyme activity can generate dierent metabolic equilibria. This mechanism can control metabolic adaptations to environmental changes and is based on the interaction between metabolic species and transcription factors which modulate enzyme biosynthesis.

Key elements in our analysis are the use of a PA model for gene regula-tion and exploiting the time scale separaregula-tion between metabolic and genetic dynamics. Applying a quasi steady state approximation ultimately permits to study the n-species gene circuit by a 2-dimensional PA system dened in conic domains. The algebraic equations in (7) will be a good approximation of the metabolite trajectories provided that the time constants of the enzyme kinetics are much smaller than protein half-lifes, which in turn are inversely proportional to their degradation rates γi. In addition, enzyme saturation may invalidate the

Regulation of Metabolic Networks 15

QSS approximation when a positive solution of (7) fails to exist. This issue has been briey mentioned, and will be addressed in an upcoming paper.

The analysis of the reduced PA system with Filippov's method [5] allows the derivation of geometric conditions on the protein expression and degradation rates to identify the behavior of solutions at the switching domain. We have obtained two angle conditions to check the existence of stable or unstable sliding motion (Lemma 4.4), and the existence of singular equilibria at the switching domains (Lemma 4.5). By combining these two results we can characterize the equilibria for a given combination of gene regulatory parameters. Because of length constraints we have limited the analysis to the setup of Lemma 5.1 (see Figure 2), but the procedure can be applied to any combination of activation or repression feedback loops.

References

[1] Uri Alon. An introduction to systems biology: design principles of biological circuits. Chapman & Hall/CRC, 2006.

[2] Richard Casey, Hidde de Jong, and Jean-Luc Gouzé. Piecewise-linear mod-els of genetic regulatory networks: equilibria and their stability. J. Math. Biol., 52:2756, 2006.

[3] M. Chaves, E.D. Sontag, and R. Albert. Methods of robustness analysis for boolean models of gene control networks. IEE Proc. Syst. Biol., 153:154 167, 2006.

[4] A. Cornish-Bowden. Fundamentals of Enzyme Kinetics. Portland Press, third edition, 2004.

[5] Aleksei F. Filippov. Dierential equations with discontinuous righthand-sides. Kluwer Academics Publishers, 1988.

[6] Leon Glass and Stuart A. Kauman. The logical analysis of continuous, nonlinear biochemical control networks. J. Theor. Biol., 39:103129, 1973. [7] Jean-Luc Gouzé and Tewk Sari. A class of piecewise linear dierential equations arising in biological models. Dynamical Systems, 17(4):299316, 2002.

[8] L.C.G.J.M. Habets and Jan van Schuppen. A control problem for ane dynamical systems on a full-dimensional polytope. Automatica, 40:2135, 2004.

[9] Oliver Kotte, Judith B Zaugg, and Matthias Heinemann. Bacterial adapta-tion through distributed sensing of metabolic uxes. Mol Syst Biol, 6(355), 2010.

[10] D. Ropers, H. de Jong, M. Page, D. Schneider, and J. Geiselmann. Qual-itative simulation of the carbon starvation response in Escherichia coli. BioSystems, 84:124152, 2006.

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